Question

A variable plane is at a distance of $6$ units from the origin. If it meets the coordinate axes in $A, B$ and $C$, then the equation of the locus of the centroid of the $\Delta A B C$ is

• A. $\frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{1}{z^{2}}=\frac{1}{4}$
• B. $x^{2}+y^{2}+z^{2}=4$
• C. $\frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{1}{z^{2}}=1$
• D. $\frac{1}{x^{2}}+\frac{1}{y^{2}}-\frac{1}{z^{2}}=\frac{1}{4}$

Answer 1

Answer: (A)

Let the equation of plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$
Distance from origin is $6 .$
$\therefore 6=\frac{1}{\sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}}}$
$ \Rightarrow \frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}=\frac{1}{36}\,\,\,...(i)$
Centroid of plane is $\left(\frac{a}{3}, \frac{b}{3}, \frac{c}{3}\right)$
$\therefore $ Let $x=\frac{a}{3} $
$\Rightarrow a=3 x$
Similarly, $b=3 y, c=3 z$
On putting the value of $a, b, c$ in Eq. (i), we get
$\frac{1}{9 x^{2}}+\frac{1}{9 y^{2}}+\frac{1}{9 z^{2}}=\frac{1}{36} $
$\Rightarrow \frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{1}{z^{2}}=\frac{1}{4}$